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Thick Difference Sets of Haar Null Compact Sets in Locally Compact Groups

Chuck Akemann

Abstract

Let \(G\) be a non-discrete, locally compact group with Haar measure \(m\). We prove that there exists a compact set \(K \subset G\) with \(m(K)=0\) such that \(KK^{-1}\) contains a neighborhood of the identity. Moreover, such a set may be constructed inside any prescribed neighborhood of the identity.

Thick Difference Sets of Haar Null Compact Sets in Locally Compact Groups

Abstract

Let be a non-discrete, locally compact group with Haar measure . We prove that there exists a compact set with \(m(K)=0\) such that contains a neighborhood of the identity. Moreover, such a set may be constructed inside any prescribed neighborhood of the identity.

Paper Structure

This paper contains 5 sections, 5 theorems, 24 equations.

Table of Contents

  1. Introduction
  2. Main result
  3. Lie case
  4. Profinite case
  5. Proof of the main theorem

Key Result

Theorem 1

Let $G$ be a non-discrete, locally compact group with Haar measure $m$, and let $O$ be an open neighborhood of the identity. Then there exists a compact set $K \subset O$ such that

Theorems & Definitions (10)

  • Theorem 1
  • Lemma 1: Pullback of null sets
  • proof
  • Lemma 2
  • proof
  • Lemma 3: Finite difference bases
  • proof
  • Lemma 4
  • proof
  • proof